What is (a+b+c) Squared?

S3-SA1-0757

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

(a+b+c) squared means multiplying the expression (a+b+c) by itself. It expands to a^2 + b^2 + c^2 + 2ab + 2bc + 2ca. This formula helps us quickly find the square of a sum of three different numbers or terms without doing long multiplication.

Simple Example

Quick Example

Imagine you have three friends, A, B, and C, who scored runs in a cricket match: A scored 10 runs, B scored 5 runs, and C scored 2 runs. If we wanted to calculate (A+B+C) squared, it's (10+5+2) squared, which is (17) squared or 289. The formula helps us break this down if the numbers were variables.

Worked Example

Step-by-Step

Let's find the value of (x+y+z)^2 when x=2, y=3, z=1.
---Step 1: Write down the formula: (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.
---Step 2: Substitute the values of a, b, and c with x, y, and z: (x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx.
---Step 3: Substitute the given numbers into the formula: (2)^2 + (3)^2 + (1)^2 + 2(2)(3) + 2(3)(1) + 2(1)(2).
---Step 4: Calculate the squares: 4 + 9 + 1.
---Step 5: Calculate the products: 2(2)(3) = 12; 2(3)(1) = 6; 2(1)(2) = 4.
---Step 6: Add all the results: 4 + 9 + 1 + 12 + 6 + 4.
---Step 7: Sum them up: 36.
So, (2+3+1)^2 = 36.

Why It Matters

Understanding how to expand (a+b+c) squared is super useful in many fields! Engineers use it to calculate areas and volumes in construction, and data scientists apply similar logic when working with large datasets to find patterns. It's a foundational concept for careers in science, technology, engineering, and even finance.

Common Mistakes

MISTAKE: Students often forget the '2ab', '2bc', '2ca' terms and only write a^2 + b^2 + c^2. | CORRECTION: Remember that squaring means multiplying the entire expression by itself, which creates cross-product terms. Always include all six terms: a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.

MISTAKE: Incorrectly calculating the '2ab' type terms, especially with negative numbers. For example, thinking 2 * (-2) * 3 is 12. | CORRECTION: Pay close attention to the signs of the numbers when multiplying. 2 * (-2) * 3 = -12.

MISTAKE: Confusing (a+b+c)^2 with a^2 + b^2 + c^2. | CORRECTION: These are not the same! (a+b+c)^2 is much larger than a^2 + b^2 + c^2 because it includes all the 'cross-multiplication' terms like 2ab, 2bc, and 2ca.

Practice Questions

Try It Yourself

QUESTION: Expand (p+q+r)^2. | ANSWER: p^2 + q^2 + r^2 + 2pq + 2qr + 2rp

QUESTION: Find the value of (2x+y+3z)^2. | ANSWER: 4x^2 + y^2 + 9z^2 + 4xy + 6yz + 12zx

QUESTION: If a=1, b=2, c=3, what is the value of (a+b+c)^2? | ANSWER: (1+2+3)^2 = 6^2 = 36

MCQ

Quick Quiz

Which of the following is the correct expansion of (x+y+z)^2?

x^2 + y^2 + z^2

x^2 + y^2 + z^2 + 2xy + 2yz + 2zx

x^2 + y^2 + z^2 + xy + yz + zx

x^2 + y^2 + z^2 + 3xyz

The Correct Answer Is:

The correct expansion includes the square of each term (x^2, y^2, z^2) and twice the product of each pair of terms (2xy, 2yz, 2zx). Options A, C, and D are missing these crucial 'cross-product' terms or have incorrect coefficients.

Real World Connection

In the Real World

Imagine you are an architect designing a building. You might use this concept to calculate the total area of a complex shape by breaking it into smaller sections (a, b, c) and then squaring their sum to find the total space. Or, in computer science, similar algebraic expansions are used in algorithms for image processing or even in financial modeling to predict market trends.

Key Vocabulary

Key Terms

EXPANSION: Writing out a mathematical expression in a longer, equivalent form by removing brackets. | TERM: A single number, variable, or a product of numbers and variables. | COEFFICIENT: A number multiplied by a variable in an algebraic term. | SQUARE: To multiply a number or expression by itself.

What's Next

What to Learn Next

Great job understanding (a+b+c) squared! Next, you can explore algebraic identities involving subtraction, like (a-b-c) squared, or even (a+b+c+d) squared. This will help you tackle more complex algebraic problems and build a strong foundation for higher mathematics.